Extension Fields: Is F(\alpha) Contain All Zeros of irr(\alpha,F)?

[ehrenfest]

[SOLVED] extension fields

Homework Statement

Let F be a field and let E be an extension field of F. Let \alpha be an element of E that is algebraic over E. Is it true that all of the zeros of irr(\alpha,F) are contained in the extension field F(\alpha)?
EDIT: I mean algebraic over F

Homework Equations

The Attempt at a Solution

 

[StatusX]

Do you mean algebraic over F? And what is irr(\alpha,F)?

 

[ehrenfest]

Yes. See the EDIT. irr(\alpha,F) is just the monic irreducible polynomial in F[x] that alpha is a zero of.

 

[morphism]

What are your thoughts about this? Have you tried to come up with a counterexample?

 

[ehrenfest]

It is false. Take (x^2+2)(x^2-2) over the rationals. Then zeros of the first factor are imaginary and the zeros of the second factor are real.

 

[morphism]

But (x^2+2)(x^2-2) is not irreducible over Q.

 

[ehrenfest]

Take x^3+3. It is irreducible by Eisenstein with p = 3. Its zeros are: \sqrt[3]{3}e^{ik\pi/3} where k=0,1,2. Two of those values of k produce imaginary roots.

 

[morphism]

That's better!